A class function on the Torelli group
Let \(\Sigma_{g,1}\) be a compact orientable surface of genus \(g\) with one boundary component. The Torelli group of \(\Sigma_{g,1}\) is defined to be the subgroup of the mapping class group of \(\Sigma_{g,1}\) acting trivially on the first integral homology \(H=H_1(\Sigma_{g,1};{\mathbb{Z}})\). The Magnus representation of \(r_1\) is a map from the Torelli group to the group \(GL(2g,{\mathbb{Z}}[H])\) defined by using the Fox derivative on the integral group ring of the fundamental group of \(\Sigma_{g,1}\), a free group of rank \(2g\). The paper under review investigates some properties of the characteristic polynomials of the images of elements of the Torelli group under the map \(r_1\).
scientific article; zbMATH DE number 2091928